A benchmark problem to evaluate implementational issues for three-dimensional flows of incompressible fluids subject to slip boundary conditions

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A benchmark problem to evaluate implementational

issues for three-dimensional flows of incompressible

fluids subject to slip boundary conditions

Radomir Chabiniok, Jaroslav Hron, Alena Jarolímová, Josef Málek,

Kumbakonam Rajagopal, Keshava Rajagopal, Helena Švihlová, Karel Tůma

To cite this version:

Radomir Chabiniok, Jaroslav Hron, Alena Jarolímová, Josef Málek, Kumbakonam Rajagopal, et

al.. A benchmark problem to evaluate implementational issues for three-dimensional flows of

in-compressible fluids subject to slip boundary conditions. Applications in Engineering Science, 2021, 6,

�10.1016/j.apples.2021.100038�. �hal-03178710�

ContentslistsavailableatScienceDirect

Applications

in

Engineering

Science

journalhomepage:www.elsevier.com/locate/apples

A

benchmark

problem

to

evaluate

implementational

issues

for

three-dimensional

flows

of

incompressible

fluids

subject

to

slip

boundary

conditions

R.

Chabiniok

_,

_J.

_Hron

_,

_A.

_Jarolímová

_,

_J.

_Málek

_,

_K.R.

_Rajagopal

_,

_K.

_Rajagopal

_,

H.

Š

vihlová

_,

_K.

_T

_ů

_ma

a Inria Saclay Ile-de-France, 1 Rue Honoré d’Estienne d’Orves, Palaiseau 91120, France b LMS, Ecole Polytechnique, CNRS, Institut Polytechnique de Paris, France

c Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Czech Republic d Department of Pediatrics, Division of Pediatric Cardiology, UT Southwestern Medical Center, Dallas, TX, United States

e Charles University, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovská 83, Prague 186 75, Czech Republic f Texas A&M University, Department of Mechanical Engineering, College Station, TX 77843-3132, United States

g University of Houston College of Medicine and Houston Heart / HCA Houston Healthcare, Houston, Texas, United States

a

b

s

t

r

a

c

t

WeconsiderflowsofanincompressibleNavier–StokesfluidinatubulardomainwithNavier’sslipboundaryconditionimposedontheimpermeablewall.Wefocus onseveralimplementationalissuesassociatedwiththistypeofboundaryconditionswithintheframeworkofthestandardTaylor-Hoodmixedfiniteelementmethod andpresentthecomputationalresultsforflowsinatubulardomainoffinitelengthwithoneinletandoneoutlet.Inparticular,wepresentthedetailsregarding variantsoftheNitschemethodconcerningtheincorporationoftheimpermeabilityconditiononthewall.Wealsofindthatthemannerinwhichthenormalto theboundaryisnumericallyimplementedinfluencesthenatureofthecomputationalresults.Asabenchmark,wesetupsteadyflowsinatubeoffinitelengthand comparethecomputationalresultswiththeanalyticalsolutions.Finally,weidentifyvariousquantitiesofinterest,suchasthedissipation,wallshearstress,vorticity, pressuredrop,andprovidetheirprecisemathematicaldefinitions.Wedocumenthowwellthesequantitiesarecomputationallyapproximatedinthecaseofthe benchmark.

Althoughthegeometryofthebenchmarkissimple,thecorrectcomputationalresultsrequirecarefulselectionofnumericalmethodsandsurprisinglynon-trivial computationalresources.Ourgoalistotest,usingthesettingwithaknownanalyticalsolution,arobustcomputationaltoolthatwouldbesuitableforcomputations onrealcomplexgeometriesthathaverelevancetoproblemsinengineeringandmedicine.Themodelparametersinourcomputationsarechosenbasedonflowsin largearteries.

1. Introduction

Mostfluidsthatcanbe describedbytheNavier–Stokesmodelare assumedtomeettheno-slipboundarycondition,thatisadherenceof thefluidtotheboundary,inflowsthattakeplaceinpipes,channels, andothersimpledomainswhentheflowis reasonablyslow.The ap-plicationofthisparticularboundaryconditionisattributedtoStokes whois supposedtohaveadvocateditsuse.Itisnotreallyclearthat Stokesbelievedinitsgeneralvalidity.Heevenhaddoubtsconcerning itsaccuracyinpipesandchannels,asheremarked“DuBuatfoundby experimentthatwhenthemeanvelocityofwaterflowingthroughapipeis lessthanaboutoneinchinasecond,thewaterneartheinnersurfaceofthe pipeisatrest.Iftheseexperimentsmaybetrusted,theconditionstobe

sat-∗_{Correspondingauthor.Theorderoftheauthorsisalphabetical.}

E-mailaddresses: radomir.chabiniok@inria.fr(R.Chabiniok), jaroslav.hron@mff.cuni.cz(J.Hron), josef.malek@mff.cuni.cz(J.Málek), krajagopal@tamu.edu

(K.R.Rajagopal),keshava.rajagopal@gmail.com(K.Rajagopal),helena.svihlova@mff.cuni.cz(H.Švihlová),ktuma@karlin.mff.cuni.cz(K.Tůma).

isfiedinthecaseofsmallvelocitiesarethosewhichfirstoccurredtome,...” TheabovecommentishardlyarousingendorsementasStokesseemsto holdtheopinionthattheno-slipboundaryconditionisonlyappropriate forreasonablyslowflows.Moreover,Stokes(1845)alsoremarks“Ihave saidthatwhenthevelocityisnotverysmallthetangentialforcecalledinto actionbytheslidingofwaterovertheinnersurfaceofapipevariesnearly asthesquareofthevelocity”,statingquiteclearlythatslipistakingplace attheboundarywhentheflowsarenotsufficientlyslow.

Boundaryconditionsreflectthemutualinteractionbetweenthefluid flowinginsidethetubeandthematerialtheboundaryismadeof.For example,forflowsinbloodvessels,oneshouldtakeintoaccountthe propertiesofbloodaswellasoftheinnerpartofthevesselwall;the properboundaryconditionshoulddescribemutualinteractionbetween

https://doi.org/10.1016/j.apples.2021.100038

Received5December2020;Receivedinrevisedform22January2021;Accepted7February2021 Availableonline11February2021

thesematerials.Thisisdefinitelyaformidabletask.Formaterialssuch asblood(whichisacomplexmixtureofplasma1_{,redandwhiteblood} cells,platelets,lipo-proteins,andavarietyofcomplexions)andblood vessel(whichhasaverycomplexlayeredstructure)itisfarfromclear thattheno-slipboundaryconditionoughttohold,see(Rajagopaland Rajagopal,2020)formoredetails.

ExperimentsconcerningtheflowofbloodincapillariesbyBennett (1967)andBugliarelloandHayden(1963)report theslippingof red bloodcellsthatcomeintocontactwiththeboundary.InNubar(1967),

Nubar(1971),MisraandShit(2007),thepossibleconnectionofslippage onthewallandshearratedependentbloodviscositymeasuredinthe rheometerissuggested.Theaboveflowsarereasonablyslowflowsand eveninsuchflowscertaincomponentsofbloodseemtobeslippingat theboundary.Castingfurtherdoubtconcerningtheappropriatenessof theno-slipboundaryconditionisthefactthattheflowofbloodcanbe turbulentatsomelocationsinthecardiovascularsystemandfarfrom theslownesspresumedbyStokeswhentheno-slipboundarycondition isexperimentallyobserved.Thus,itwouldbeworthwhiletostudythe flowofamateriallikebloodbyassumingtheslipboundarycondition.

Numeroustypesofboundaryconditionsthatgobeyondtheno-slip boundarycondition wereproposed bytheearlypioneersinthefield of fluiddynamics (deCoulomb,1800; DuBuat,1786; Girard, 1813; Navier,1823;deProny,1804),andPoisson(1831). See(Netoetal., 2005)foramorerecentoverviewofexperimentalstudiesonboundary slipmeasurements.

Inthisstudy,wefocusonNavier’sslipboundarycondition(proposed byNavier,1823inagreementwithmoleculararguments)characterized bytheparameter𝜃 andinvolving,asthelimitingcases,no-slip bound-aryconditionatoneend(𝜃 =1)and(perfect)slipboundarycondition ontheotherend(𝜃 =0).Thisextendedclassofboundaryconditions, takingintoaccounttheslippingmechanismsofvariousdegree, repre-sentsthe most commonlyusedboundaryconditions imposedon the impermeablepartsoftheboundaryoftheflowdomain.Interestingly, itwasshown(see(Hronetal.,2008))thatforbothplaneand three-dimensionalPoiseuilleflowvaryingtheslipparameter𝜃 caninfluence theflowmoremarkedlythanthechangeoftheconstitutiveequationfor thefluid.

WeconsiderflowsofanincompressibleNavier–Stokesfluidina spe-cialthree-dimensionaldomain,namelyatubulardomainoffinitelength withoneinletandoneoutlettogetherwiththeabovedescribedclassof slipboundaryconditionsimposedontheimpermeablewall.Thereason forchoosingthisspecialgeometricalsetupistwofold.First,knowingthe analyticalsolutionsforsteadyflowssubjecttodifferentslippageatthe wallofcylinder,weareabletosetabenchmarkthatcanserveasabasic testforvariousnumericalmethodsandtheirimplementations.Second, thisgeometricalsettingcanbeviewedasaverysimplified,yet reason-ableapproximationofflowsinalargevessels,applicableforinstance wheninvestigating/assessingflowthroughaorticvalve(nativeorits re-placement),orapathologicallynarrowedsegmentoflargeartery(e.g. coarctationofaortaorpulmonaryarterystenosis).

Next,wepresentenergeticconsiderationsthatmightprovidesupport forthechoiceofboundaryconditionsthatweenforceatarigid bound-aryandattheoutletandprovidethefunctionspacesinwhichtheweak formulationsoftheproblemisformulated.Theseformulationsdiffer de-pendingonhowtheincompressibilityconstraintandtheimpermeability conditionareincorporated.Inparticular,wepresentthedetails regard-ingvariantsoftheNitschemethod(Nitsche,1971)concerning(inthis study)theincorporationoftheimpermeabilityconditiononthewall. WeprovidealiteratureoverviewconcerningtheNitschemethodatthe endofSection3.

Then,wefocusonseveralissuesassociatedwiththeimplementation ofthistypeofboundaryconditionswithintheframeworkofthe

stan-1 _{PlasmaisoftentimesdescribedbyaNavier–Stokesfluidwhileflowingin}

largebloodvessels.

Fig.1. Computationaldomainandthepartsoftheboundary.

dardTaylor-Hoodfiniteelements.Wediscusshowthemannerinwhich thenormaltotheboundaryisnumericallyimplementedinfluencesthe natureofthecomputationalresults.Wealsoidentifyvariousquantities ofinterest,suchasthedissipation,pressuredrop,vorticity,wallshear stress,andprovidetheirprecisemathematicaldefinitions.Theobjective istodocumenthowwellthesequantitiesarecomputationally approxi-matedinthecaseoftheproposedbenchmark.

Wewouldliketoemphasizethatalthoughthegeometryofthe bench-markissimple,thecorrectcomputationalresultsrequireacareful selec-tionofnumericalmethodsandnon-trivialcomputationalresources.Our goalistotest,usingthesettingwithaknownanalyticalsolution,a ro-bustcomputationaltoolthatwouldbesuitableforcomputationsonreal complexgeometriesthathaverelevancetoproblemsinengineeringand medicine.Themodelparametersinourcomputationsarechosenbased onflowsinlargearteries;thiscorrespondstoflowsatReynoldsnumber ofapproximately1050.

Thestructureofthepaperisthefollowing.InSection2.1,we formu-latetheproblemdescribingunsteadyflowsofincompressibleNavier– Stokesfluidinthree-dimensionaltubewithNavier’sslipboundary con-ditionontheimpermeablewall,agivenvelocityattheinletandspecial boundaryconditionsattheoutletthattakesintoaccountthegiven nor-malstress(i.e.thepressure)aswellasthecontrol(boundedness)ofthe energyofthewholesystem.Theenergyconsiderationsalsoleadtothe choiceofthefunctionspacesinwhichwestatetheweakformulation oftheproblem.InSection3,wetakeaslightlydifferentviewpointwith regardtotheweakformulationoftheproblemandwediscusstwo vari-antsoftheNitschemethodthatareusedtonumericallyincorporate,in aweaksense,theimpermeabilitycondition.Section4providesabrief descriptionofthenumericaldiscretizationofweakformulations con-nectedwiththeNitschemethod.Theemphasisisdevotedtothewayin whichthenormalvectorisimplemented.InSection5,thebenchmark withknownanalyticalsolutionunderthesimplifyingassumptionofthe simpleshearflowisformulated.Wealsodefinetwoformsofmechanical dissipationofthesystemandpresentitsconnectiontothekineticenergy andthepressuredropalongthelengthofthetube.Pressuredropisoften usedinlieuofthedissipationrateincaseswheretherealdissipationis unknown.ThisisclarifiedinSection5.2.Section6bringstogetherall theresultsconnectedwithournumericalapproachandcomputational toolsandcomparesthemwiththeanalytical(benchmark)solution.We alsocomparetheefficiencyofthevariantsoftheNitschemethod.The concludingSection7summarizesthemainresultsthathavebeen estab-lished.

2. Definitionoftheproblemandenergyconsideration

2.1. Formulationoftheinitial-boundary-valueproblem

WeconsidertheflowofanincompressibleNavier–Stokesfluidover timeinterval𝑇,inathree-dimensionaltube ofthefinitelength rep-resentingabounded,openandconnectedsetΩ⊂ 𝐑3_{,seeFig.1. The}

unsteady flows of suchfluids described in termsof the velocity𝐯= (𝑣1,𝑣2,𝑣3)∶(0,𝑇)× Ω→ 𝐑3 andthe mean normalstress −𝑝∶(0,𝑇)×

followingsetofequations: div𝐯=0 in(0,𝑇)× Ω, (1) 𝜌∗𝜕𝐯_𝜕𝑡 +𝜌∗(∇𝐯)𝐯=div𝕋 in(0,𝑇)× Ω, (2) 𝕋=−𝑝𝕀+𝜇∗ ( ∇𝐯+∇𝐯𝑇) in(0,𝑇)× Ω, (3) where𝕋istheCauchystressand𝜌∗>0,𝜇∗>0aregivenconstants:the

densityandthedynamicviscosity.Wealsousethenotation

[(∇𝐯)𝐯]_𝑖= 3 ∑ 𝑘=1 𝜕𝑣𝑖 𝜕𝑥𝑘𝑣𝑘= 3 ∑ 𝑘=1 𝜕 𝜕𝑥𝑘(𝑣𝑘𝑣𝑖)=∶[div(𝐯⊗ 𝐯)], 𝑖=1,2,3, wheretheincompressibilityconstraint(1)hasbeenincorporated. Fur-thermore,foranymatrix𝔸,thesymbol𝔸𝑇 _{denotesitstranspose.}

Theboundary𝜕Ω ofthetubulardomainΩconsistsofthree non-overlappingpartsΓin,ΓoutandΓwallandweimposethefollowinginitial

andboundaryconditionsinvolvingthreegivenfunctions:𝐯0∶Ω→ 𝐑3,

𝐯in:[0,𝑇]× Γin→ 𝐑3and𝑃:[0,𝑇]→ 𝐑: 𝐯(0,⋅)=𝐯0 inΩ, (4) 𝐯=𝐯in on(0,𝑇)× Γin, (5) 𝕋𝐧=−𝑃(𝑡)𝐧+1 2𝜌∗(𝐯⋅ 𝐧)−𝐯 on(0,𝑇)× Γout, (6) 𝐯⋅ 𝐧=0 and 𝜃𝐯𝜏+𝛾∗(1−𝜃)(𝕋𝐧)𝜏=𝟎 on(0,𝑇)× Γwall. (7)

Here𝜃 ∈ [0,1],𝛾∗∈ (0,∞)and𝐧isaunitnormalvectorattheboundary.

Furthermore,for𝑥∈𝐑weset𝑥+∶=max{0,𝑥}and𝑥−∶=min{0,𝑥},i.e.

𝑥=𝑥++𝑥−,andforanarbitraryvector𝐳,𝐳𝐧∶=(𝐳⋅ 𝐧)𝐧isthenormal

componentof𝐳 and𝐳𝜏∶=𝐳−𝐳𝐧 istheprojectionof 𝐳 tothetangent

plane.

Notethatinsteadof(6)wecouldimposethecondition

𝐯𝜏=𝟎 and 𝕋𝐧⋅ 𝐧=−𝑃(𝑡)+1 2𝜌∗(𝐯⋅ 𝐧)

2

− on(0,𝑇)× Γout. (8)

Bytakingthe scalarproductof (6)with𝐧,we observethat(6)and

(8)givetheidenticaldescriptioninthenormaldirection.Ontheother hand,byprojecting(6)intothetangentplane,weobtain1

2𝜌∗(𝐯⋅ 𝐧)−𝐯𝜏=

(𝕋𝐧)_𝜏,whichisdifferentthanthefirstconditionin(8).Thecondition

(8)ismoreappropriatetothesituationswherethevelocityfieldatthe outflowis unidirectional(whichis,infact,thecaseoftheanalytical solutionusedinSection5).Althoughtheboundaryconditions(8)are differentthanthosestatedin(6),inthefewcomputationaltestsmade sofarwehavenotobservedanysignificantdifference.Thisiswhythe conditions(8)arenotdiscussedanymorebelowandweconsidermerely

(6)atΓoutinwhatfollows.

Theboundaryconditions(6),(7)andthefunctionspacesfortheweak formulationareinkeepingwithbasicenergyestimatesthatcanbe es-tablishedfortheproblem(1)–(7),seebelow.Wewishtoremarkthat thepresenceoftheterm1₂𝜌∗(𝐯⋅ 𝐧)−𝐯intheoutflowboundarycondition (6)seems tobe essentialnotonlyforobtainingtheenergyestimates (seebelow) butalsofortherobustnessandstabilityofthenumerical methods.

2.2. Energybalanceandenergyestimates

Forclarity,weproceedintwosteps.First,wesimplifythesettingby assumingthat

𝐯in=𝟎. (9)

Ageneralinflowfunctionwillbeconsideredlater.

Toobtaintheenergyidentity,assuming(9),weformthescalar prod-uctof(2)with𝐯,use(1)andarriveat

𝜕 𝜕𝑡 ( 𝜌∗|𝐯| 2 2 ) +div ( 𝜌∗|𝐯| 2 2 𝐯−𝕋𝐯 ) +𝕋⋅ 𝔻(𝐯)=0, (10)

where𝔻(𝐯)isthesymmetric partof thevelocitygradientdefined as

1 2

(

∇𝐯+∇𝐯𝑇).ThenweintegratethisequationoverΩ,applytheGauss theorem, usetheimpermeabilityconditionon Γwall andthe

homoge-neousDirichletconditiononΓinandconcludethat

d d𝑡∫Ω𝜌∗ |𝐯|2 2 d𝑥+∫Ω𝕋⋅ 𝔻 (𝐯)d𝑥+ ∫Γ_out𝜌∗ |𝐯|2 2 𝐯⋅ 𝐧d𝑆 + ∫Γ_wall∪Γ_out −𝕋𝐯⋅ 𝐧d𝑆=0. (11)

Duetothesymmetryof𝕋,wehave𝕋𝐯⋅ 𝐧=𝕋𝐧⋅ 𝐯,andbydecomposing thevectors𝕋𝐧and𝐯intotheirnormalcomponentandprojectioninto thetangentplaneandusingtheimpermeabilityconditiononΓwallwe

concludefrom(11)that

d d𝑡∫Ω𝜌∗ |𝐯|2 2 d𝑥+∫Ω𝕋⋅ 𝔻 (𝐯)d𝑥+ ∫Γwall −(𝕋𝐧)_𝜏⋅ 𝐯_𝜏d𝑆 + ∫Γout 𝜌∗|𝐯| 2 2 (𝐯⋅ 𝐧)+d𝑆=∫Γout ( 𝕋𝐧⋅ 𝐯−𝜌∗|𝐯| 2 2 (𝐯⋅ 𝐧)− ) d𝑆. (12) Thisstructureisconsistentwiththeboundaryconditions(6)and(7)to beconsideredinthisstudy.Takingintoaccounttheincompressibility condition(1),(12)canbeexpressedas(seeBraackandMucha,2014)

d d𝑡∫Ω𝜌∗ |𝐯|2 2 d𝑥+∫Ω𝕋⋅ 𝔻 (𝐯)d𝑥+ ∫Γ_wall −(𝕋𝐧)_𝜏⋅ 𝐯_𝜏d𝑆 + ∫Γout 𝜌∗|𝐯| 2 2 (𝐯⋅ 𝐧)+d𝑆 =− ∫Γout 𝑃(𝑡)𝐯⋅ 𝐧d𝑆=−𝑃(𝑡) ∫_𝜕Ω𝐯⋅ 𝐧d𝑆=−𝑃(𝑡)∫Ω div𝐯d𝑥=0. (13)

Asthefluidisincompressiblethereisanindeterminateparttothestress thatisspherical.Setting−𝑝∶=1₃(𝕋11+𝕋22+𝕋33),i.e.themeannormal

stress, wehave𝕋 =−𝑝𝕀+𝕋_𝑑,where𝕋𝑑∶=𝕋−13(𝕋11+𝕋22+𝕋33)𝕀.

Ne-glecting, forsimplicity,thelast (non-negative)termon theleft-hand sideof(13)andapplyingtheintegrationw.r.t.timeover(0,𝑡),𝑡∈ (0,𝑇], weget ∫Ω𝜌∗ |𝐯(𝑡,𝑥)|2 2 d𝑥+∫ 𝑡 0 ∫Ω𝕋𝑑⋅ 𝔻 (𝐯)d𝑥d𝜏 + ∫ 𝑡 0 ∫Γwall −(𝕋𝐧)_𝜏⋅ 𝐯_𝜏d𝑆d𝜏 ≤_∫ Ω𝜌∗ |𝐯0(𝑥)|2 2 d𝑥. (14)

Theaboveequationstatesthatthetotalkineticenergyattime𝑡andthe sumofthedissipatedenergyduetointernalfrictioninthefluid consid-eredover(0,𝑡)× Ωandthedissipatedenergyduetofrictioncausedby interactionofthefluidwiththeinnerpartoftheboundaryiscontrolled (bounded)bytheinitialkineticenergy.Inordertobeconsistentwith thesecondlawofthermodynamics,thetotaldissipatedenergyhastobe non-negative.This isforexampleautomaticallyfulfilled byrequiring that

𝕋𝑑=2𝜇∗𝔻(𝐯) on(0,𝑇)× Ω, (15)

−(𝕋𝐧)_𝜏= 1

̃𝛾∗𝐯𝜏

on(0,𝑇)× Γwall, (16)

where𝜇∗and̃𝛾∗arepositiveconstantsrepresentingthedynamic

viscos-ityandthefrictioncoefficient,respectively.WenotethatEqs.(15)and

(16)areconstitutiverelationsandarelinear.Inordertoincorporatea parameterthatmeasuresthevariousdegreesofslippageandincludes

alsono-slip(𝐯𝜏=𝟎)and(perfect)slip((𝕋𝐧)𝜏=𝟎)boundaryconditions weintroducetheequation

𝜃𝐯𝜏+𝛾∗(1−𝜃)(𝕋𝐧)𝜏=𝟎 on(0,𝑇)× Γwall, (17)

where0≤𝜃 ≤1insteadof(16).Toconclude,uponinserting(15)and

(17)into(14),weobtainthefollowingenergyinequalityfor𝜃 ∈ (0,1): ∫Ω𝜌∗ |𝐯(𝑡,𝑥)|2 2 d𝑥+2𝜇∗∫ 𝑡 0 ∫Ω|𝔻 (𝐯)|2_d𝑥d𝜏 + 𝜃 𝛾∗(1−𝜃)∫ 𝑡 0 ∫Γwall |𝐯𝜏|2d𝑆d𝜏 ≤ ∫Ω𝜌∗ |𝐯0(𝑥)|2 2 d𝑥. (18)

For𝜃 =0or𝜃 =1theenergyinequalitytakesasimplerform,namely ∫Ω𝜌∗ |𝐯(𝑡,𝑥)|2 2 d𝑥+2𝜇∗∫ 𝑡 0 ∫Ω|𝔻 (𝐯)|2d𝑥d𝜏 ≤ ∫Ω𝜌∗ |𝐯0(𝑥)|2 2 d𝑥. (19) Next,weconsiderthecompleteformulation(1)–(7),i.e.withgeneral inflowvelocity𝐯in.Weassumetheexistenceofanextension𝐯∗inof𝐯in,i.e.

werequirethatthereexistsacontinuousfunction𝐯∗

in∶(0,𝑇)× Ω→ 𝐑 3_,

ΩbeingtheclosureofΩ,suchthat div𝐯∗_in=0 in (0,𝑇)× Ω, 𝐯∗ in=𝐯in onΓin, 𝐯∗ in⋅ 𝐧=0 on Γwall, 𝐯∗ in=𝟎 on Γout, 𝜕𝐯∗ in 𝜕𝑡 ∈𝐿2((0,𝑇)× Ω)3 and∇𝐯∗in∈𝐿 ∞_{((0,𝑇)× Ω)}3×3_. (20)

Toobtaintheenergyequality,wemodifytheaboveprocedureandform thescalarproductof(2)with𝐯−𝐯∗_in.Doingso,weobtain

𝜕 𝜕𝑡 ( 𝜌∗ |𝐯−𝐯∗_in|2 2 ) +div ( 𝜌∗ |𝐯−𝐯∗_in|2 2 𝐯 ) −div(𝕋(𝐯−𝐯∗_in))+𝕋⋅ 𝔻(𝐯−𝐯∗_in) =−𝜌∗ 𝜕𝐯∗ in 𝜕𝑡 ⋅ (𝐯−𝐯∗in)−𝜌∗𝑣𝑘 𝜕𝐯∗ in 𝜕𝑥𝑘 ⋅ (𝐯−𝐯 ∗ in) =−𝜌∗ 𝜕𝐯∗ in 𝜕𝑡 ⋅ (𝐯−𝐯∗in)−𝜌∗(𝐯−𝐯∗in)⊗ (𝐯−𝐯 ∗ in)⋅ ∇𝐯 ∗ in −𝜌∗((𝐯−𝐯∗in)⊗ 𝐯 ∗ in)⋅ ∇𝐯 ∗ in =−𝜌∗ 𝜕𝐯∗ in 𝜕𝑡 ⋅ (𝐯−𝐯∗in)−𝜌∗(𝐯−𝐯∗in)⊗ (𝐯−𝐯 ∗ in)⋅ ∇𝐯 ∗ in +𝜌∗(𝐯∗in⊗ 𝐯 ∗ in)⋅ 𝔻(𝐯−𝐯 ∗ in)−div ( 𝜌∗(𝐯∗in⋅ (𝐯−𝐯 ∗ in))𝐯 ∗ in ) . (21)

ThenweintegratethisequationoverΩanduseGauss’stheorem.Using alsothepropertiesof𝐯∗

ingivenin(20),weobtain d d𝑡∫Ω𝜌∗ |𝐯−𝐯∗ in| 2 2 d𝑥+∫Ω𝕋⋅ 𝔻 (𝐯−𝐯∗_in)d𝑥+ ∫Γwall −(𝕋𝐧)_𝜏⋅ (𝐯_𝜏−(𝐯∗_in)_𝜏)d𝑆 + ∫Γout 𝜌∗|𝐯| 2 2 (𝐯⋅ 𝐧)+d𝑆 =− ∫Ω 𝜌∗ 𝜕𝐯∗ in 𝜕𝑡 ⋅ (𝐯−𝐯∗in)d𝑥−_∫ Ω 𝜌∗(𝐯−𝐯∗in)⊗ (𝐯−𝐯 ∗ in)⋅ ∇𝐯 ∗ ind𝑥 + ∫Ω𝜌∗ (𝐯∗_in⊗ 𝐯∗_in)⋅ 𝔻(𝐯−𝐯∗_in)d𝑥−𝑃(𝑡) ∫Γout 𝐯⋅ 𝐧d𝑆. (22) Uponinsertingtheconstitutiveequations(15)and(17)(noticingthat theboundaryintegralonΓwallvanishesfor𝜃 =0and𝜃 =1priorto

in-serting)andneglectingthelast(non-negative)termontheright-hand sideof(22)wearriveat d d𝑡∫Ω𝜌∗ |𝐯−𝐯∗ in| 2 2 d𝑥+2𝜇∗∫Ω|𝔻 (𝐯−𝐯∗_in)|2d𝑥 + 𝜃 𝛾∗(1−𝜃) { ∫Γwall |𝐯𝜏|2d𝑆−_∫ Γwall 𝐯𝜏⋅ (𝐯∗in)𝜏d𝑆 } ≤− ∫Ω𝜌∗ (𝐯−𝐯∗ in)⊗ (𝐯−𝐯 ∗ in)⋅ ∇𝐯 ∗ ind𝑥+_∫ Ω𝜌∗ (𝐯∗ in⊗ 𝐯 ∗ in)⋅ 𝔻(𝐯−𝐯 ∗ in)d𝑥 − ∫Ω𝜌∗ 𝜕𝐯∗ in 𝜕𝑡 ⋅ (𝐯−𝐯∗in)d𝑥+𝑃(𝑡)_∫ Γin 𝐯⋅ 𝐧d𝑆 −2𝜇∗_∫ Ω 𝔻(𝐯∗ in)⋅ 𝔻(𝐯−𝐯 ∗ in)d𝑥, (23)

wherethelast termhasbeenaddedtotheleft-handside.Next, inte-grating(23)w.r.t.timeandusingthestandardnotationforthenorms intheLebesguespaces(𝐿𝑝(Ω),‖ ⋅ ‖_𝑝)for1≤𝑝≤∞,andusingalsothe standardCauchy–Schwartz,YoungandHölderinequalities,weobtain

1 2𝜌∗‖(𝐯−𝐯 ∗ in)(𝑡,⋅)‖ 2 2+2𝜇∗_∫ 𝑡 0 ‖𝔻 (𝐯−𝐯∗_in)‖2₂d𝑠 + 𝜃 𝛾∗(1−𝜃)∫ 𝑡 0 ∫Γwall |𝐯𝜏|2_d𝑆d𝑠 ≤ 1 2𝜌∗‖𝐯0−𝐯 ∗ in(0,⋅)‖ 2 2+ 3𝜇∗ 2 ∫ 𝑡 0 ‖𝔻(𝐯−𝐯∗ in)‖ 2 2d𝑠 +𝐶(𝐯∗_in)𝜌∗_∫ 𝑡 0 ‖𝐯 −𝐯∗_in‖2₂d𝑠+𝑃(𝑡) ∫ 𝑡 0 ∫Γin 𝐯in⋅ 𝐧d𝑆d𝑠 +1 2 𝜃 𝛾∗(1−𝜃)∫ 𝑡 0 ∫Γwall (|𝐯_𝜏|2+|(𝐯∗_in)𝜏|2)d𝑆d𝑠+𝐶(𝐯∗_in), (24)

where the constant 𝐶(𝐯∗

in) depends on 𝜌∗, 𝜇∗, ∫0𝑇‖‖‖‖ 𝜕𝐯∗ in 𝜕𝑡 ‖‖‖‖ 2 2 d𝑠, ∫0𝑇‖𝔻(𝐯 ∗ in)‖ 2

2d𝑠 and sup𝑡∈(0,𝑇 )‖∇𝐯∗in‖∞ and is finite due to the

as-sumptionson𝐯∗

instatedin(20).

Finally,movingthesecondtermfromtheright-handsidetotheleft andapplyingtheGronwalllemmaweconcludethat

𝜌∗ sup 𝑡∈[0,𝑇 ]‖(𝐯−𝐯 ∗ in)(𝑡,⋅)‖ 2 2+𝜇∗_∫ 𝑇 0 ‖𝔻 (𝐯−𝐯∗_in)‖2₂d𝑡 + 𝜃 𝛾∗(1−𝜃)∫ 𝑇 0 ∫Γwall |𝐯𝜏|2d𝑆d𝑡 ≤𝐶(𝑇,‖𝐯0−𝐯∗in(0,⋅)‖ 2 2,𝐶(𝐯 ∗ in) ) <∞. (25)

2.3. Definitionofweaksolutiontotheproblem(1)–(7)

Let 𝑊1,2 div, bc∶= { 𝜙𝜙𝜙 ∈ 𝑊1,2_(Ω)3_{; div𝜙𝜙𝜙 =0,𝜙𝜙𝜙 =𝟎onΓ} in,𝜙𝜙𝜙 ⋅ 𝐧=0onΓwall } .

Motivatedbytheaboveenergyestimates,bytheconceptofLeray-Hopf weaksolutionfortheNavier–Stokesequationswithno-slipboundary conditions,andbytheresultsconcerningthe𝐿𝑝_{-maximalregularityfor} theevolutionaryStokessystem,we saythatacouple(𝐯,𝑝)is aweak solutionto(1)–(7)providedthat

𝐯−𝐯∗ in ∈ 𝐿 2_(0,𝑇;𝑊1,2 div, bc)∩𝐶𝑤𝑒𝑎𝑘([0,𝑇];𝐿 2_(Ω)3_), 𝜕𝐯 𝜕𝑡 ∈ 𝐿5∕4(0,𝑇;𝐿5∕4(Ω)), ₍₂₆₎ 𝑝∈ 𝐿5∕4_(0,𝑇;𝑊1,5∕4_(Ω)), lim 𝑡→0+‖𝐯(𝑡,⋅)−𝐯0‖2→ 0, and ∫Ω 𝜌∗𝜕𝐯_𝜕𝑡⋅ 𝜙𝜙𝜙 d𝑥+_∫ Ω 𝜌∗(∇𝐯)𝐯⋅ 𝜙𝜙𝜙 d𝑥+2𝜇∗_∫ Ω 𝔻(𝐯)⋅ 𝔻(𝜙𝜙𝜙)d𝑥 + 𝜃 𝛾∗(1−𝜃)∫Γ_wall𝐯𝜏⋅ 𝜙𝜙𝜙𝛕 d𝑆 =−𝑃(𝑡) ∫Γout 𝜙𝜙𝜙 ⋅ 𝐧d𝑆+ ∫Γout 𝜌∗ 2(𝐯⋅ 𝐧)−𝐯⋅ 𝜙𝜙𝜙 d𝑆 validforall𝜙𝜙𝜙 ∈ 𝑊1,2

Alternatively,insteadof(27),onecanconsider ∫Ω𝜌∗ 𝜕𝐯 𝜕𝑡⋅ 𝜙𝜙𝜙 d𝑥−∫Ω𝜌∗ (𝐯⊗ 𝐯)⋅ ∇𝜙𝜙𝜙 d𝑥+2𝜇∗_∫ Ω𝔻 (𝐯)⋅ 𝔻(𝜙𝜙𝜙)d𝑥 + 𝜃 𝛾∗(1−𝜃)∫Γwall 𝐯𝜏⋅ 𝜙𝜙𝜙𝛕d𝑆+_∫ Γout 𝜌∗(𝐯⋅ 𝐧)𝐯⋅ 𝜙𝜙𝜙 d𝑆 =−𝑃(𝑡) ∫Γout 𝜙𝜙𝜙 ⋅ 𝐧d𝑆+ ∫Γout 𝜌∗ 2(𝐯⋅ 𝐧)−𝐯⋅ 𝜙𝜙𝜙 d𝑆 validforall𝜙𝜙𝜙 ∈ 𝑊1,2

div, bcanda.a.𝑡∈ (0,𝑇). (28)

3. TheNitschemethod

Inthissectionwemodifytheweakformulation(27)sothatthe in-compressibilityconstraint(1)andtheimpermeabilitycondition(7)1on

thewallaretreatedinamannersuitablefornumericalimplementation. Thus,thefunctionspace𝑊1,2

div, bc introducedintheprevioussectionis modifiedinsuchawaythatittakesintoaccountonlytheinflow bound-arycondition,i.e.insteadofusing𝑊1,2

div, bcthevelocity𝐯isnowsought inthespace𝑉 suchthat

𝑉 ∶={𝐯∈𝑊1,2(Ω)3,𝐯=𝟎onΓin}. (29)

Thenweproceedasfollows.First,theincompressibilityconstraint

(1)ismultipliedbythetestfunction𝑞∈𝐿2_{(Ω)andintegratedoverΩ.}

Then,thebalanceoflinearmomentum(2)ismultipliedbythetest func-tion𝜙𝜙𝜙 ∈ 𝑉 integratedoverΩfollowedbytheintegrationbypartswith respecttotheterminvolving−div𝕋.Next,theimpermeability condi-tionwrittenwithrespecttotheform−𝐯𝐯𝐯_𝐧=𝟎onΓwallismultipliedby

thetestfunction𝜓𝜓𝜓∈𝐿2_(Γ

wall)3 andintegratedoverΓwall. Finally,we

sumtheestablishedidentitiesandobtain

∫Ω ( 𝜌∗𝜕𝐯_𝜕𝑡+𝜌∗(∇𝐯)𝐯 ) ⋅ 𝜙𝜙𝜙 d𝑥+ ∫Ω𝕋 (𝐯,𝑝)⋅ ∇𝜙𝜙𝜙 d𝑥+ ∫Ω (div𝐯)𝑞d𝑥 + ∫Γwall 𝜃 𝛾∗(1−𝜃)𝐯𝜏⋅ 𝜙𝜙𝜙𝜏 d𝑆− ∫Γwall (𝕋(𝐯,𝑝)𝐧)_𝐧⋅ 𝜙𝜙𝜙_𝐧d𝑆− ∫Γwall 𝐯𝐧⋅ 𝜓𝜓𝜓d𝑆 − ∫Γout ( −𝑃(𝑡)𝐧+1 2𝜌∗(𝐯⋅ 𝐧)−𝐯 ) ⋅ 𝜙𝜙𝜙 d𝑆=0

forall(𝑞,𝜙𝜙𝜙,𝜓𝜓𝜓)∈ (𝐿2(Ω)×𝑉×𝐿2(Γwall)3), (30)

whereweusethenotation𝕋(𝜙𝜙𝜙,𝑞)∶=−𝑞𝕀+2𝜇𝔻(𝜙𝜙𝜙)andwherewealso usedtheobservationthat

∫_𝜕Ω𝕋𝐧⋅ 𝜙𝜙𝜙 d𝑆=∫Γout ( −𝑃(𝑡)𝐧+1 2𝜌∗(𝐯⋅ 𝐧)−𝐯 ) ⋅ 𝜙𝜙𝜙 d𝑆 − ∫Γwall 𝜃 𝛾∗(1−𝜃)𝐯𝜏⋅ 𝜙𝜙𝜙𝜏 d𝑆+ ∫Γwall (𝕋𝐧)_𝐧⋅ 𝜙𝜙𝜙_𝐧d𝑆. (31)

Theformulation(30)requiresspecialtestfunctions𝜓𝜓𝜓 definedonthe boundaryΓwallwhichoneneedstoenlargetheproblem.Itistempting

toreplacethisnewtestfunctionbyacombinationof𝑞 and𝜙𝜙𝜙,whichis possibleas𝜓𝜓𝜓actson𝐯whichisalreadytheunknown.

Inspiredbythestructureofthelast twotermsin themiddleline of(30)weusethetestfunction𝜓𝜓𝜓=(𝕋(𝜙𝜙𝜙,𝑞)𝐧)_𝐧andobtaintheoriginal Nitschemethod.Nitsche(1971)showedthatsuchaformulationisnot stable,butwhenoneaddsthestabilizationterm

𝛽 ℎ∫Γwall

𝐯𝐧⋅ 𝜙𝜙𝜙𝐧d𝑆, 𝛽 >0 (32)

theweakformbecomesstableforsufficientlylargeparameter𝛽 >0.The localsizeofmeshedgeisdenotedbyℎ.Sincetheextraterm−∫_Γ

wall𝐯𝐧⋅

(𝕋(𝜙𝜙𝜙,𝑞)𝐧)_𝐧d𝑆 hasthesamesign astheterm−∫_Γ

wall(𝕋(𝐯,𝑝)𝐧)𝐧⋅ 𝜙𝜙𝜙𝐧d𝑆,

thismethodiscalledthesymmetricNitschemethod.Theweakformulation toourproblemusingthesymmetricNitschemethodthenreads:

Find(𝐯−𝐯∗_in,𝑝)∈𝐿∞(0,𝑇;𝐿2(Ω)3)∩𝐿2(0,𝑇;𝑉)×𝐿5∕4(0,𝑇;𝐿2(Ω)) suchthat ∫Ω ( 𝜌∗𝜕𝐯_𝜕𝑡 +𝜌∗(∇𝐯)𝐯 ) ⋅ 𝜙𝜙𝜙 d𝑥+2𝜇∗_∫ Ω𝔻 (𝐯)⋅ 𝔻(𝜙𝜙𝜙)d𝑥 − ∫Ω𝑝 (div𝜙𝜙𝜙)d𝑥+ ∫Ω (div𝐯)𝑞d𝑥− ∫Γout ( −𝑃(𝑡)𝐧+1 2𝜌∗(𝐯⋅ 𝐧)−𝐯 ) ⋅ 𝜙𝜙𝜙 d𝑆 + ∫Γwall 𝜃 𝛾∗(1−𝜃)𝐯𝜏⋅ 𝜙𝜙𝜙𝜏 d𝑆− ∫Γwall (𝕋(𝐯,𝑝)𝐧)_𝐧⋅ 𝜙𝜙𝜙_𝐧d𝑆 − ∫Γwall 𝐯𝐧⋅ (𝕋(𝜙𝜙𝜙,𝑞)𝐧)_𝐧d𝑆+𝛽 ℎ∫Γwall 𝐯𝐧⋅ 𝜙𝜙𝜙𝐧d𝑆=0

forall (𝝓,𝑞)∈𝑉×𝐿2(Ω)anda.a.𝑡>0. (33)

Themaindisadvantageofaddingstabilization(32)isthatthe pa-rameter 𝛽 isproblem-dependent andithastobe manuallyadjusted. Later,Burman(2012)showedthattheoriginalNitschemethodcanbe improved(inthesenseofdropping𝛽)bychangingthesigninfrontof theboundaryintegral

∫Γwall

𝐯𝐧⋅ (𝕋(𝜙𝜙𝜙,𝑞)𝐧)_𝐧d𝑆, (34) whichcorresponds totaking𝜓𝜓𝜓=−(𝕋(𝜙𝜙𝜙,𝑞)𝐧)_𝐧in(31).Thismethodis calledthenon-symmetricNitschemethodanditsweakformulationreads:

Find(𝐯−𝐯∗_in,𝑝)∈𝐿∞(0,𝑇;𝐿2(Ω)3)∩𝐿2(0,𝑇;𝑉)×𝐿5∕4(0,𝑇;𝐿2(Ω)) suchthat ∫Ω ( 𝜌∗𝜕𝐯_𝜕𝑡 +𝜌∗(∇𝐯)𝐯 ) ⋅ 𝜙𝜙𝜙 d𝑥+2𝜇∗_∫ Ω𝔻 (𝐯)⋅ 𝔻(𝜙𝜙𝜙)d𝑥 − ∫Ω 𝑝(div𝜙𝜙𝜙)d𝑥+ ∫Ω (div𝐯)𝑞d𝑥+ ∫Γwall 𝜃 𝛾∗(1−𝜃) 𝐯𝜏⋅ 𝜙𝜙𝜙𝜏d𝑆 − ∫Γ_wall (𝕋(𝐯,𝑝)𝐧)_𝐧⋅ 𝜙𝜙𝜙_𝐧d𝑆+ ∫Γ_wall𝐯𝐧⋅ (𝕋 (𝜙𝜙𝜙,𝑞)𝐧)_𝐧d𝑆 − ∫Γ_out ( −𝑃(𝑡)𝐧+1 2𝜌∗(𝐯⋅ 𝐧)−𝐯 ) ⋅ 𝜙𝜙𝜙 d𝑆=0

forall(𝝓,𝑞)∈𝑉×𝐿2(Ω)anda.a.𝑡>0. (35)

Inourcomputationalresultspresentedbelowweusebothformulations

(33)and(35).

Weconcludethissectionwithabriefliteratureoverviewregarding theNitschemethod.Afiniteelementapproximationofincompressible Navier–Stokesequationswithslipontheboundaryconditionis investi-gatedinVerfürth(1986,1991).Thesepapersprovidetherelationofthe StokesflowwithslipontheboundarytothewellknownBabuška para-dox forboundarysupportedshellplate,(see. Babuška,1959;Rieder, 1974),whereadiscreteproblemwithpiece-wiselinearboundary ap-proximationconvergestoasolutionthatisdifferentfromthe contin-uous one. Asobserved inVerfürth (1986)theshellproblem studied byBabuška isin factequivalenttoastreamfunctionformulationof Stokesflowwithslipontheboundary.Toworkaroundthisproblem theyshowthatmixedformulationwiththeLagrangemultipliermethod toimposetheslipboundaryconditionforNavier–Stokesequationsis stable andconvergestothedesired solution.InDione etal.(2013);

Urquizaetal.(2014),acomparisonofpenaltymethod,Lagrange multi-pliermethodandseveralvariantsoftheNitschemethodtoimposeslip boundaryconditiononcurvedboundaryin2Dispresentedwith obser-vationthatsomevariantsdonotexhibitconvergencepredictedbythe theory.JuntunenandStenberg(2009)derivedstabilityanderror esti-matesforthesymmetricNitschemethodappliedtoaPoissonequation

Fig. 2. Twodifferent implementations of the normalvector𝐧ℎ ontheboundary𝜕Ωℎ:

Piece-wiseconstantfacetnormals𝐧𝑓

ℎ (left)vs.

contin-uousvertexnormals𝐧𝑣 ℎ(right).

Table1

Thevaluesoftheconstantparametersusedincalculation.

Symbol Name Value Unit

𝜌∗ The density 1050 kg m −3

𝜇∗ The dynamic viscosity 3.896 ×10 −3 Pa . s = kg.m −1 s −1 𝑅 The radius of the cylinder 12 ×10 −3 _m

𝐿 The length of the cylinder 44 ×10 −3 _m 𝛽∗ The characteristic length 12 ×10 −3 m 𝛾∗ The first slip parameter 𝛽

∗

𝜇∗ 3.08 m

2 . s . kg −1 𝜃 The second slip parameter 0.0–1.0 – 𝑉 The mean inlet velocity 0.65 m . s −1

withDirichletboundarydata.InLayton(1999)itisproposedthatweak impositionofno-slipboundaryconditionissuperiortostronglyenforced conditionsandinBeckeretal.(2015)theNitschemethodisusedto ap-plygeneralboundaryconditionsforEulerandNavier–Stokesequations. Finally,Mekhloufetal.(2017)presentscomparisonofthesymmetric andnon-symmetricNitschemethodsforenforcingDirichletconditions inaweaksenseforseveralsimple2Dflowproblem.

4. Thefiniteelementdiscretization

Tosolvenumericallythevariants(33)and(35)wediscretizethe sys-temsbyTaylor-Hoodfiniteelements𝑃2∕𝑃1inspace(Boffi etal.,2013; Wieners,2003).ForthefiniteelementdiscretizationtheFEniCSlibrary (Alnæsetal.,2015)isused.SincethegivenparametersinTable1 re-quireasufficientlyfinemeshtoobtainstablesolution,weaddthe stan-dardSUPGstabilization(Boffi etal.,2013)toobtainapproximate solu-tiononcoarsemeshes.Forthetimediscretizationweusesimpleimplicit Eulermethodandperformtheiterationprocessuntilasteadystate solu-tionisreached,see

TSPSEUDO

4.1. Theimportanceofdefinitionofdiscreteboundarynormalvector

Itisclearthatthediscreteimplementationofsuchamethodcan dependonthechoiceofboundarydiscretizationandthedefinitionof normalandtangentvectorstotheboundarycaninfluencethecomputed quantities.SeeSimeandWilson(2020),Engelmanetal.(1982),Stokes andCarey(2011)forsimilardiscussion.

Togiveaninterestedreaderawarningregardingthisissuewegive onestrikingexamplebelow.Before,weneedtofixsomenotation.We approximateoursmoothdomainΩbyapiece-wiselinear(planar) poly-hedraldomainwithboundaryΩ_ℎ.Thecorrespondingdiscretenormal vector𝐧𝑓_ℎisdenotedasfacetnormal,seeFig.2,anditspiece-wise con-stantfunctiononthesurfacemesh𝜕Ωℎ.Itisthenormalprovidedbythe usedFEniCSnumericallibraryas

𝐧𝑓_ℎ=FacetNormal(mesh).

Wecanalsointroduceavertexnormalbytakingthenormalvectorin vertices,i.e.𝐧𝑣

ℎ.This canbe constructed inseveralways,see for ex-ample(Dioneetal.,2013).Wechosetodefinea vertexnormalasa

𝐿2-projectionof𝐧𝑓_ℎ tospaceofcontinuouspiece-wiselinearfunctions

on𝜕Ωℎ,seeagainFig.2.Thisisrealizedbycomputingthe𝐿2projection

as

𝐧𝑣

ℎ=arg_𝐧∈𝑁min||𝐧−𝐧𝑓ℎ||22, where𝑁={𝐧∈𝐶(𝜕Ωℎ),𝐧|𝑇∈𝑃1(𝑇) ∀𝑇∈𝜕Ωℎ}. Finally,in thecase ofthestraight pipewe canuse theradialvector scaledtounitlengthasananalyticnormaltothesurface,thisvariantis denotedby𝐧𝑎

ℎ,thatmeans

𝐧𝑎

ℎ=(𝑥,𝑦,0)∕|(𝑥,𝑦,0)|.

There are some additional possibilities with regard to defining theboundarynormal,forexampleusingthedistancefunction𝑑(𝑥)= distance(𝑥,𝜕Ω_ℎ)totheboundary𝜕Ωℎ,wehave‖ ∇𝑑‖ =1.Thismight beaprioriknownorcanbeobtainedbysolvingtheEikonalequation, oritsregularizedversion.Thenthenormalvectorcouldbeobtainedas

𝐧𝑑

ℎ=∇𝑑,however,inoursimplesituation,thisperformsalmost iden-ticallytothevertexnormal𝐧𝑣

ℎ andthis iswhywedonotreportany resultsforthisapproachhere.

Nowwereturntothepromisedexample.Letusconsiderasimple calculationofPoiseuilleflowwithfullslipboundarycondition.The an-alyticalsolutionforthisproblemis trivial:aconstantpressureanda constantvelocityfield𝐯𝑎_=(0,0,𝑣

𝑧).Fig.3showsthenumericalresults fordifferentchoicesofnormalvector.Theleftpanelshowsthe numer-icalresultifweusetheradialvector𝐧𝑎_ℎasthenormalvector.Thenwe recoverthecontinuoussolution.Howeverifweusethenormal𝐧𝑓_ℎ cor-respondingtothefacetsofthecomputationaldomainΩ_ℎ,showninthe middlepanel,theapproximationofthevelocityfieldbecomesverybad (≈ 40%error).Ontheotherhandtakingthevertexbasednormal𝐧𝑣

ℎ re-ducestheerrortoabout4%,whichisshownintherightpanel.Yetthis isaverysimpleproblem.

5. Descriptionofbenchmark

Theproblem presentedin Section2.1will benowstudiedin the regime of steadyflowsin thecylindricaldomainshown inFig.1.It meansthatforgiven𝐯in∶Γin→ 𝐑3and𝑃∈𝐑westudythefollowing

problem:tofind(𝐯,𝑝)∶Ω→ 𝐑3_{×𝐑satisfying} div𝐯=0 inΩ, (36) 𝜌∗(∇𝐯)𝐯=div𝕋with𝕋=−𝑝𝕀+𝜇∗ ( ∇𝐯+∇𝐯𝑇) inΩ, (37) 𝐯=𝐯in onΓin, (38) 𝕋𝐧=−𝑃𝐧+1 2𝜌∗(𝐯⋅ 𝐧)−𝐯 onΓout, (39) 𝐯⋅ 𝐧=0 and 𝜃𝐯_𝜏+𝛾_∗(1−𝜃)(𝕋𝐧)_𝜏=𝟎 onΓ_wall. (40) Intherestofthissection,weproceedasfollows.First,inSection5.1, starting fromtheassumptionthatthecylinderisinfinite,theflowis steadyandtakestheform ofasimpleshearflow,we derivethe an-alyticalsolutionforsteadyflowsoftheNavier–Stokesfluidsatisfying

Fig.3. Fullslipflowonaverycoarsemesh(with5650degreesoffreedom).Theanalyticalsolutionis𝐯𝑎_=(0,0,𝑣

𝑧).The3discretesolutionscorrespondtothe

symmetricNitschemethodwith3differentnormalvectors– analytic,facetandvertexnormals(𝐧𝑎

ℎ,𝐧𝑓ℎ,𝐧𝑣ℎ).Thecolorshowstherelativeerrorbetweenthediscrete

solutionandtheanalyticalone,i.e.|𝐯ℎ−𝐯 𝑎_|

|𝐯𝑎_| .

Fig.4. Cylinderdimensions.

theboundarycondition(40)onΓwall.Thissolutionwillservebothas

theinflowboundarycondition𝐯inin theproblem(1)–(7)butalsoas

abenchmarkusedfortestingthenumericalmethodsandtheir imple-mentations.Then,wewilldefinethedissipation,vorticityandpressure dropinSection5.2andcomputetheirvaluesfortheanalyticalsolution foundinSection5.1.Thedescriptionofthebenchmarkandthevalues ofparametersareprovidedinSection5.3.

5.1. Steadyflowinthecylinder

Fora steady flow in a cylinderwith radius𝑅 located along the

𝑧−axis,seeFig.4,onecanlookforthevelocityintheformofsimple shearflow,i.e.,

𝐯=(0,0,𝜔(𝑥,𝑦)). (41) Then,theincompressibilityconstraint(36)isfulfilledandthesystemof

Eqs. (37)simplifiesto 𝑝=−𝐺𝑧+𝐶, (42) 𝜇∗ ( 𝜕2_𝜔 𝜕𝑥2 + 𝜕2_𝜔 𝜕𝑦2 ) =−𝐺. (43)

Here,𝐺 hasthemeaningofthepressuregradientandcanbespecified as

𝐺= 𝑝𝑖𝑛−𝑝𝑜𝑢𝑡

𝐿 >0 (44)

where𝑝_𝑖𝑛and𝑝_𝑜𝑢𝑡 aredefinedasthepressureaverageovertwocross sectionsΓinandΓout,i.e.,

𝑝𝑖𝑛=_|Γ1

in| ∫Γin

𝑝d𝑆 and 𝑝_𝑜𝑢𝑡= 1 |Γout| ∫Γout

𝑝d𝑆. (45)

Notethatfrom(42)itfollowsthat𝑝𝑖𝑛=−12𝐺𝐿+𝐶 and𝑝𝑜𝑢𝑡= 1 2𝐺𝐿+𝐶

forthecross-sectionsΓinandΓoutlocatedat𝑧=∓𝐿∕2.Denotingfurther

themeaninflowvelocityas

𝑉 = 1 |Γin| ∫Γin

𝜔(𝑥,𝑦)d𝑆, (46)

itiswellknownthattheanalyticalsolutionsforthelimitcasesof no-slip(𝜃 =1)andoffull-slip(𝜃 =0)taketheparabolicorconstantvelocity profiles,i.e.,

𝜔(𝑥,𝑦)= 2𝑉(𝑅

2_−(𝑥2_+𝑦2₎₎

𝑅2 for𝜃 =1, (47)

𝜔(𝑥,𝑦)=𝑉 for𝜃 =0. (48)

Therefore,for𝜃 ∈ (0,1),wecanusetheansatz2

𝜔(𝑥,𝑦)=𝑉(𝐶1(1−𝜃)+𝐶2𝜃(𝑅2−(𝑥2+𝑦2))

)

(49) withgeneralparameters𝐶1,𝐶2thatmaydependon𝜃.Withthisansatz,

weobtain 𝜕𝜔 𝜕𝑥=−2𝑥𝑉𝐶2𝜃, 𝜕𝜔 𝜕𝑦 =−2𝑦𝑉𝐶2𝜃, 𝜕 2_𝜔 𝜕𝑥2 = 𝜕2_𝜔 𝜕𝑦2 =−2𝑉𝐶2𝜃 inΩ, (50) 𝕋𝐧= ⎛ ⎜ ⎜ ⎝ −𝑝 0 −2𝜇∗𝑉𝐶2𝜃𝑥 0 −𝑝 −2𝜇∗𝑉𝐶2𝜃𝑦 −2𝜇∗𝑉𝐶2𝜃𝑥 −2𝜇∗𝑉𝐶2𝜃𝑦 −𝑝 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 𝑥 𝑅_𝑦 𝑅 0 ⎞ ⎟ ⎟ ⎠, (𝕋𝐧)_𝜏 = ⎛ ⎜ ⎜ ⎝ 0 0 −2𝜇∗𝑉𝐶2𝜃𝑅 ⎞ ⎟ ⎟ ⎠ onΓwall. (51)

From(46)and(40)2wegetthefollowingrestrictions:

𝜋𝑅2_{𝑉 =} ∫ 2𝜋 𝜑=0∫ 𝑅 𝑟=0𝑉 ( 𝐶1(1−𝜃)+𝐶2𝜃(𝑅2−𝑟2) ) 𝑟d𝑟d𝜑, (52) 𝜃𝑉𝐶1(1−𝜃)=2𝛾∗(1−𝜃)𝜇∗𝑉𝐶2𝜃𝑅. (53) Thisleadsto 𝐶1= 4𝛾∗𝜇∗𝑅 𝑅[4𝛾∗𝜇∗(1−𝜃)+𝜃𝑅 ], (54) 𝐶2= 2 𝑅[4𝛾∗𝜇∗(1−𝜃)+𝜃𝑅 ], (55) 𝐺= 8𝜇∗𝑉𝜃 𝑅[4𝛾∗𝜇∗(1−𝜃)+𝜃𝑅 ]. (56)

2_{One can alternatively use the cylindrical coordinates and solve (43),}

Table2

Convergenceofrelativeerrorswithmeshrefinement-Naviersslip𝜃 =0.5,non-symmetricNitsche,analyticnormal.DOFs:numberofdegreesoffreedom;EOC: estimatedorderofconvergence;ΞΩ:bulkdissipation;ΞΓ:walldissipation;𝑝drop:pressuredifferencebetweeninletandoutlet.

DOFS ‖𝐯𝑎−𝐯ℎ‖𝐿2

‖𝐯𝑎‖𝐿2 EOC

‖𝐩𝑎−𝐩ℎ‖𝐿2

‖𝐩𝑎‖𝐿2 EOC ΞΩ ΞΓ flux pdrop

5 650 3.08 ×10 −4 _{− 4.31 ×10} −2 _{− 7.12 ×10} −2 _{2.05 ×10} −2 _{3.58 ×10} −3 _{6.15 ×10} −2

39 098 2.10 ×10 −4 _{0.55 1.28 ×10} −2 _{1.75 1.79 ×10} −2 _{6.70 ×10} −3 _{1.27 ×10} −3 _{1.73 ×10} −2

290 000 5.79 ×10 −5 _{1.86 2.26 ×10} −3 _{2.50 4.39 ×10} −3 _{1.64 ×10} −3 _{4.09 ×10} −4 _{2.23 ×10} −3

2 232 28 1.78 ×10 −5 _{1.70 7.74 ×10} −4 _{1.55 1.16 ×10} −3 _{4.02 ×10} −4 _{1.32 ×10} −4 _{6.52 ×10} −4

Table3

Convergenceofrelativeerrorswithmeshrefinement– Navier’sslip𝜃 =0.5,non-symmetricNitsche,facetnormal.

DOFS ‖𝐯𝑎−𝐯ℎ‖𝐿2

‖𝐯𝑎‖𝐿2 EOC

‖𝐩𝑎−𝐩ℎ‖𝐿2

‖𝐩𝑎‖𝐿2 EOC ΞΩ ΞΓ flux pdrop

5 650 1.63 ×10 −2 _{− 6.07 ×10} 0 _{− 1.28 ×10} −1 _{5.10 ×10} −2 _{1.03 ×10} −4 _{3.00 ×10} 0

39 098 7.46 ×10 −3 _{1.13 1.67 ×10} 0 _{1.86 1.48 ×10} −1 _{2.55 ×10} −2 _{2.58 ×10} −3 _{6.09 ×10} −1

290 000 2.35 ×10 −3 _{1.67 4.18 ×10} −1 _{2.00 6.08 ×10} −2 _{1.14 ×10} −2 _{7.02 ×10} −5 _{1.33 ×10} −1

2 232 028 6.98 ×10 −4 _{1.75 1.09 ×10} −1 _{1.94 2.11 ×10} −2 _{4.04 ×10} −3 _{6.52 ×10} −1 _{3.67 ×10} −2

Table4

Convergenceofrelativeerrorswithmeshrefinement– Navier’sslip𝜃 =0.5,non-symmetricNitsche,vertexnormal.

DOFS ‖𝐯𝑎−𝐯ℎ‖𝐿2

‖𝐯𝑎‖𝐿2 EOC

‖𝐩𝑎−𝐩ℎ‖𝐿2

‖𝐩𝑎‖𝐿2 EOC ΞΩ ΞΓ flux pdrop

5 650 1.62 ×10 −2 _{− 5.34 ×10} 0 _{− 7.25 ×10} −2 _{3.67 ×10} −2 _{2.57 ×10} −2 _{3.77 ×10} 0 39 098 7.21 ×10 −3 _{1.17 1.69 ×10} 0 _{1.66 1.21 ×10} −1 _{1.72 ×10} −2 _{5.73 ×10} −3 _{5.92 ×10} −1 290 000 2.30 ×10 −3 _{1.65 4.20 ×10} −1 _{2.01 5.07 ×10} −2 _{8.75 ×10} −3 _{4.06 ×10} −4 _{1.29 ×10} −1 2 232 028 6.89 ×10 −4 _{1.74 1.08 ×10} −1 _{1.96 1.73 ×10} −2 _{3.18 ×10} −3 _{7.13 ×10} −5 _{3.75 ×10} −2 Consequently, 𝜔(𝑥,𝑦)=𝑉4𝛾∗𝜇∗𝑅(1−𝜃)+2𝜃(𝑅 2_−(𝑥2_+𝑦2₎₎ 𝑅[4𝛾∗𝜇∗(1−𝜃)+𝜃𝑅 ] , (57) or,using𝛽∗=𝛾∗𝜇∗, 𝜔(𝑥,𝑦)=𝑉4𝛽∗𝑅(1−𝜃)+2𝜃(𝑅 2_−(𝑥2_+𝑦2₎₎ 𝑅[4𝛽∗(1−𝜃)+𝜃𝑅 ] . (58) 5.2. Dissipation

Toobtaintheformulaforthetotal dissipationin thecaseof the evolutionaryproblemdescribed by(1)–(7), westartwiththeenergy identity,see(10),andintegrateitoverΩ.UsingtheGausstheorem,the formof𝕋andthefactthat𝐯⋅ 𝐧=0onΓwallweobtain

d dt∫Ω ( 𝜌∗ |𝐯|2 2 ) dx+ ∫Ω 2𝜇_∗|𝔻(𝐯)|2_dx+ ∫Γin∪Γout 𝜌∗ |𝐯|2 2 (𝐯⋅ 𝐧)dS − ∫𝜕Ω𝕋𝐯⋅ 𝐧d𝑆=0. (59)

Duetosymmetryof𝕋 andtheboundaryconditions,thelasttermleads to − ∫𝜕Ω𝕋𝐧⋅ 𝐯d𝑆 = ∫Γ_in𝑝 (𝐯in⋅ 𝐧)d𝑆+_∫ Γ_out𝑃 (𝑡)(𝐯⋅ 𝐧)d𝑆− ∫Γ_in 2𝜇∗𝔻(𝐯)⋅ 𝐧d𝑆 − ∫Γout 𝜌∗|𝐯| 2 2 (𝐯⋅ 𝐧)−d𝑆−∫Γwall ( (𝕋𝐧)_𝑛+(𝕋𝐧)_𝜏 ) ⋅(𝐯𝑛+𝐯𝜏)d𝑆 = ∫Γin (𝑝−𝑃(𝑡))(𝐯in⋅ 𝐧)d𝑆−_∫ Γin 2𝜇∗𝔻(𝐯)⋅ 𝐧d𝑆 − ∫Γout 𝜌∗|𝐯| 2 2 (𝐯⋅ 𝐧)−d𝑆+ 𝜃 𝛾∗(1−𝜃)∫Γwall 𝐯𝜏⋅ 𝐯𝜏d𝑆, (60) whereweused thefactthat∫Γout(𝐯⋅ 𝐧)=−∫Γin(𝐯in⋅ 𝐧),which follows

fromdiv𝐯=0,theGausstheoremandtheimpermeabilityconditionon

thewall.Combining(59)and (60)weobtain d d𝑡𝐸k(𝑡)+ΞΩ(𝑡)+𝐽𝑝(𝑡)+ΞΓ(𝑡)+𝐽d,in(𝑡)+𝐽k,in(𝑡)+𝐽k,out(𝑡)=0, (61) where 𝐸k(𝑡):=_∫ Ω𝜌∗ |𝐯|2 2 d𝑥≥0, (62) ΞΩ(𝑡):=_∫ Ω 2𝜇∗|𝔻(𝐯)|2d𝑥≥0, (63) ΞΓ(𝑡):= 𝜃 𝛾∗(1−𝜃)∫Γwall |𝐯𝜏|2d𝑆≥0, (64) 𝐽d,in(𝑡):=−_∫ Γin 2𝜇∗𝔻(𝐯)𝐧⋅ 𝐯ind𝑆, (65) 𝐽𝑝(𝑡):=_∫ Γin (𝑝−𝑃(𝑡))𝐯in⋅ 𝐧d𝑆≤0, (66) 𝐽k,in(𝑡):=_∫ Γin 𝜌∗ |𝐯in|2 2 ( 𝐯in⋅ 𝐧 ) d𝑆≤0, (67) 𝐽k,out(𝑡):=_∫ Γout 𝜌∗|𝐯| 2 2 (𝐯⋅ 𝐧)+d𝑆≥0. (68) ThequantitiesΞΩandΞΓ representbulkandwalldissipation,

respec-tively,and𝐽_𝑝(𝑡)isthefluxcorrespondingtothepressuredrop.The quan-tity𝐽d,inisthefluxovertheinletcomingfromthediffusionterm,while

𝐽k,in,𝐽k,outarethefluxesovertheinletandoutletgeneratedbythe

con-vectiveterm.

Finally,welookatthesimplificationof theidentity(61)andthe correspondingquantitiesinthecaseoftheanalyticalsolutionforsteady simpleshearflowsderivedintheprevioussubsection.Wefirstobserve that d

d𝑡𝐸k=0.Moreover,as𝐯=𝐯in=(0,0,𝜔)where𝜔=𝜔(𝑥,𝑦)≥0we

Table5

Convergenceofrelativeerrorswithmeshrefinement– Navier’sslip𝜃 =0.5,symmetricNitsche,analyticnormal.

DOFS ‖𝐯𝑎−𝐯ℎ‖𝐿2

‖𝐯𝑎‖𝐿2 EOC

‖𝐩𝑎−𝐩ℎ‖𝐿2

‖𝐩𝑎‖𝐿2 EOC ΞΩ ΞΓ flux 𝑝 drop

5 650 1.43 ×10 −3 _{− 3.76 ×10} −1 _{− 6.44 ×10} −2 _{1.65 ×10} −2 _{9.17 ×10} −4 _{4.26 ×10} −1

39 098 3.75 ×10 −4 _{1.93 6.72 ×10} −2 _{2.48 1.58 ×10} −2 _{5.81 ×10} −3 _{2.47 ×10} −4 _{8.06 ×10} −2

290 000 7.56 ×10 −5 _{2.31 9.68 ×10} −3 _{2.80 3.90 ×10} −3 _{1.47 ×10} −3 _{3.98 ×10} −5 _{1.07 ×10} −2

2 232 028 1.95 ×10 −5 _{1.95 2.00 ×10} −3 _{2.27 1.07 ×10} −3 _{3.73 ×10} −4 _{6.29 ×10} −6 _{1.65 ×10} −3

Table6

Convergenceofrelativeerrorswithmeshrefinement– Navier’sslip𝜃 =0.5,symmetricNitsche,facetnormal.

DOFS ‖𝐯𝑎−𝐯ℎ‖𝐿2

‖𝐯𝑎‖𝐿2 EOC

‖𝐩𝑎−𝐩ℎ‖𝐿2

‖𝐩𝑎‖𝐿2 EOC ΞΩ ΞΓ flux 𝑝 drop

5 650 1.09 ×10 −1 _{− 3.13 ×10} 1 _{− 7.24 ×10} 0 _{4.47 ×10} −1 _{1.45 ×10} −2 _{3.39 ×10} 1

39 098 9.18 ×10 −2 _{0.25 2.01 ×10} 1 _{0.64 8.95 ×10} 0 _{4.84 ×10} −1 _{3.60 ×10} −3 _{2.04 ×10} 1

290 000 4.68 ×10 −2 _{0.97 9.06 ×10} 0 _{1.15 4.86 ×10} 0 _{3.23 ×10} −1 _{1.28 ×10} −3 _{8.98 ×10} 0

2 232 028 2.15 ×10 −2 _{1.12 3.95 ×10} 0 _{1.20 2.04 ×10} 0 _{1.75 ×10} −1 _{4.04 ×10} −4 _{3.86 ×10} 0

Table7

Convergenceofrelativeerrorswithmeshrefinement– Navier’sslip𝜃 =0.5,symmetricNitsche,vertexnormal.

DOFS ‖𝐯𝑎−𝐯ℎ‖𝐿2

‖𝐯𝑎‖𝐿2 EOC

‖𝐩𝑎−𝐩ℎ‖𝐿2

‖𝐩𝑎‖𝐿2 EOC ΞΩ ΞΓ flux 𝑝 drop

5 650 5.97 ×10 −3 _{− 2.78 ×10} 0 _{− 9.12 ×10} −2 _{3.05 ×10} −2 _{2.44 ×10} −3 _{1.21 ×10} 0 39 098 6.32 ×10 −3 _{− 0.08 1.54 ×10} 0 _{0.85 8.42 ×10} −2 _{4.57 ×10} −3 _{2.04 ×10} −3 _{1.03 ×10} 0 290 000 1.70 ×10 −3 _{1.89 4.30 ×10} −1 _{1.84 2.93 ×10} −2 _{2.02 ×10} −4 _{5.33 ×10} −4 _{2.28 ×10} −1 2 232 028 4.51 ×10 −4 _{1.91 1.07 ×10} −1 _{2.01 8.06 ×10} −3 _{2.10 ×10} −4 _{1.23 ×10} −4 _{3.72 ×10} −2 Table8 Relativeerrors|𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐−𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑

𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐 | betweenanalyticandcomputedvariablesfortwomeshes(c0=coarse,c1=fine)fornon-symmetricNitschemethod.Ifanalyticvalue

=0,thevalueofcomputedresultsisshown.

𝜃 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ΞΩ[J/s] c0 0.00 135.65 19.55 26.59 21.75 16.18 11.38 7.43 4.21 1.52 1.25 c1 0.00 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 ΞΓ[J/s] c0 0.00 5.98 5.99 6.00 6.02 6.05 6.11 6.22 6.47 7.17 0.00 c1 0.00 0.06 0.06 0.06 0.06 0.05 0.05 0.04 0.03 0.00 0.00 Ξ[J/s] c0 0.00 9.48 4.48 2.85 2.05 1.61 1.34 1.19 1.13 1.15 1.25 c1 0.00 0.06 0.06 0.06 0.05 0.05 0.05 0.04 0.03 0.03 0.04 1 |Ω|‖rot 𝐯 ‖1 [1/s] c0 2.92 11.16 22.14 15.98 10.55 6.92 4.40 2.56 1.17 0.08 1.03 c1 0.00 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.32 0.34 𝐽 _{𝑝 [J/s]} c0 -0.00 8.20 19.54 27.01 28.98 28.22 25.41 20.45 12.59 0.23 19.95 c1 0.00 0.11 0.11 0.11 0.11 0.11 0.10 0.09 0.07 0.03 0.88 𝑝 drop [mmHg] c0 0.00 8.22 19.53 27.00 28.97 28.21 25.40 20.45 12.59 0.24 19.95 c1 -0.00 0.10 0.10 0.10 0.10 0.10 0.09 0.08 0.06 0.03 0.88 1 |Γwall|‖( 𝕋 𝐧 ) 𝜏‖1,Γwall [ Pa ] c0 0.03 35.93 2.88 2.76 3.52 2.78 1.34 0.33 1.83 3.13 5.25 c1 0.00 0.02 0.03 0.03 0.03 0.03 0.03 0.03 0.02 0.01 0.01

wholedomain,weget𝐽d,in=0.Finally,𝑝=𝑝𝑖𝑛=12𝐺𝐿+𝐶=𝐺𝐿+𝑃(𝑡)

onΓin.Therefore,(61)simplifiesto ΞΩ+ΞΓ+𝐽𝑝=0⇒ Ξ ∶=ΞΩ+ΞΓ=−𝐽𝑝. (69) Furthermore, |𝔻(𝐯)|2= 8𝑉 2_𝜃2_(𝑥2_+𝑦2₎ 𝑅2[_4𝛽 ∗(1−𝜃)+𝜃𝑅 ]2 inΩ, (70) |𝐯𝜏|2₌ 16𝑉 2_𝛽2 ∗(1−𝜃)2 [ 4𝛽∗(1−𝜃)+𝜃𝑅 ]2 onΓwall, (71)

andwegetthefollowingexplicitformulaforthedissipationand pres-suredrop,namely

ΞΩ[J∕s]=2𝜇∗_∫ 𝐿 𝑧=0∫ 2𝜋 𝜑=0∫ 𝑅 𝑟=0 8𝜃2_𝑉2_𝑟2 𝑅2_[4𝛽 ∗(1−𝜃)+𝜃𝑅]2 𝑟d𝑟d𝜑d𝑧 = 8𝜋𝜃 2_𝑉2_𝑅2_𝐿𝜇 ∗ [4𝛽∗(1−𝜃)+𝜃𝑅]2 , (72) ΞΓ[J∕s]= _𝛾 𝜃 ∗(1−𝜃)∫ 𝐿 𝑧=0∫ 2𝜋 𝜑=0 16𝑉2_𝛽2 ∗(1−𝜃)2 [ 4𝛽∗(1−𝜃)+𝜃𝑅 ]2𝑅d𝜑d𝑧 = 32𝑉 2_𝜇2 ∗𝛾∗𝜋𝑅𝐿𝜃(1−𝜃) [ 4𝛽∗(1−𝜃)+𝜃𝑅 ]2 , (73)

Fig.5.Convergencecomparison,Navier’sslipwith𝜃 =0.5-differentnormals:

Fig.6.Convergencecomparison,Navier’sslipwith𝜃 =0.5-differentnormals:

Ξ[J∕s]=ΞΩ+ΞΓ= 8𝜋𝜃𝑉2_𝑅𝐿𝜇 ∗ [ 4𝛽∗(1−𝜃)+𝜃𝑅 ] , (74) 𝐽𝑝[J∕s]=𝐺𝐿_∫ Γ_in𝐯⋅ 𝐧 d𝑆=−𝐺𝐿𝜋𝑅2_{𝑉, (75)} 𝑝drop[Pa]=𝑝𝑖𝑛−𝑝𝑜𝑢𝑡=𝐺𝐿= 8𝜃𝑉𝐿𝜇∗ 𝑅[4𝛽∗(1−𝜃)+𝜃𝑅 ]. (76)

Moreover,wedefinethe𝐿1_{-normofthevorticityby}

‖rot𝐯‖1[m3∕s]=_∫ Ω || || | ( 𝜕𝜔 𝜕𝑦,−𝜕𝜔𝜕𝑥,0 )_|| || |d𝑥= 8𝑉𝜃|Ω| 3[4𝛽∗(1−𝜃)+𝜃𝑅 ] = 8𝜋𝑅 2_𝐿𝑉𝜃 3[4𝛽∗(1−𝜃)+𝜃𝑅 ] (77)

Fig.7. ComparisonoftheanalyticalandcomputeddissipationsΞΩ,ΞΓ,Ξ,pressuredrop𝑝drop,pressureflux𝐽𝑝and𝐿1normofvorticitydividedbyvolumeand𝐿1

normofwallshearstressdividedbywallareafortubulargeometrycomparedtotheanalyticalsolutionwithbenchmarkparametersdefinedinSection5.3.

andthe𝐿1_{-normofthewallshearstressby}

‖(𝕋𝐧)_𝜏‖1,Γwall[Pa.m 2_]= ∫Γwall |(𝕋𝐧)_𝜏| d𝑆=_[ 4𝜇∗𝑉𝜃|Γwall| 4𝛽∗(1−𝜃)+𝜃𝑅 ] = [ 8𝜋𝜇∗𝑉𝜃𝑅𝐿 4𝛽∗(1−𝜃)+𝜃𝑅 ]. (78) 5.3. Benchmarkparameters

Thegeometricalandflowparameterswerechosentobeascloseas possibletothoseforbloodflowintheaorticrootorasegmentof tho-racicaorta (motivatedbytheassessmentofclinical significanceof a

stenosedaorticvalve).Thegeometryconsideredisthatofa12mm ra-dius(24mmdiameter)cylinder.Thisisconsistentwithtypicalaortic di-mensions.Thegeometricaldatatogetherwiththedensity,thedynamic viscosityandtheinflowvelocityweretakenfromŠvihlová etal.(2017). ForgeometricaldataseealsoMadukauwa-Davidetal.(2018).

Therearetwoslipparametersprescribedonthewall,𝛾∗and𝜃,and

theyarerelatedtoeachother,see(40).Weusetheparameter𝛽∗=𝛾∗𝜇∗

asacharacteristiclengthofthegeometry,specifiedasradiusoftheinlet

𝑅.ThevaluesoftheconstantsandtheirunitsareshowninTable1.These valuesresultintheReynoldsnumber𝑅𝑒=𝑉𝑅𝜌∗

2𝜇∗

=1051.Twoprescribed functionsare𝐯in:=(0,0,𝜔(𝑥,𝑦))with𝜔(𝑥,𝑦)givenby(58)andtheoutlet

Fig.8. ComparisonoftheanalyticalandcomputeddissipationsΞΩ,ΞΓ,Ξ,pressuredrop𝑝drop,pressureflux𝐽𝑝,𝐿1normofvorticitydividedbyvolumeand𝐿1norm

ofwallshearstressdividedbywallareawithregardtotheparameter𝜃 fordifferent𝛾∗.Thesetwoparametersarerelatedtoeachotherthrough(7).

6. Results

Nextwecanlookatthenumericalresultsforthemodeldescribed in Section 2.1 as compared to the analytical solutions derived in

Section5.2.Thisgivesustheopportunitytoassesseveralaspectsofthe discretizationmethod.Firstofall,themeshconvergencecanbe com-paredforthesymmetricNitscheformulationwithdifferentpenalization parameters𝛽 andthenon-symmetricNitscheformulation.Then,witha selectedformulationwecomparethequantitiesofinterestforthefull scaleofboundaryconditionsfromfullslip(𝜃 =0.0)tonoslip(𝜃 =1.0).

6.1. Symmetricvs.non-symmetricNitscheformulation

Tables2–7summarizethebehaviorofthediscretesolutionwith re-specttothemeshrefinement.Thecomputationswerecarriedoutonthe sequenceof4regularlyrefinedmeshes.Fig.5collectstheconvergence behaviorforrelativeerrors ofvelocity𝐯andpressure𝑝in𝐿2 _norms

as ‖𝐯𝑎−𝐯ℎ‖𝐿2

‖𝐯𝑎‖𝐿2 and

‖𝐩𝑎−𝐩_ℎ‖_𝐿2

‖𝐩𝑎‖𝐿2 .Each graphcompares thebehaviorofthe

errorforthe3variantsofdiscretenormalvectors.Itisclearthatthe

analyticnormalgivesthemostaccuratesolution,howeverthisis avail-ableonlyinspecialcasesandcannotbeusedinageneralgeometry. Behaviorforthetwoothernormalsnowdependsonthevariantofthe Nitschemethodused.Thenon-symmetricNitschemethodgivesalmost thesameresultsforbothnormals– facetbasedandvertexbased.Inthe caseofsymmetricNitschemethodthevertexbasednormalgiveserrors comparablewiththenon-symmetricmethodbutthefacetbasednormal resultsinmuchlargererrorsandifthepenaltyparameterisnotchosen correctlyitseemstoconvergeatalowerrateofconvergence.

Fig.6portraysthesamecomparisonforthedissipation.Here the observationissimilar– namelythatthecombinationofthesymmetric Nitschemethodwiththefacetbasednormalleadstomuchlargererrors andslowerconvergencecompared toalltheothercombinations.We note thattheselectionof thevalueforthepenaltyparameter𝛽 was

givenbythefactthatforsmallervaluestheconvergenceisnotachieved, probablyduetolesserstability,andforhigherpenaltyparameterthe resultingaccuracydegrades.

Thefinalobservationisthattheresultsforthenon-symmetric vari-antoftheNitschemethodarenotsensitivetothechoiceofthenormal